Annals of Statistics

Statistical Estimation and Optimal Recovery

David L. Donoho

Full-text: Open access

Abstract

New formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random Gaussian noise. The formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class. It is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules, for a variety of loss functions. It is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated. The method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery.

Article information

Source
Ann. Statist., Volume 22, Number 1 (1994), 238-270.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325367

Digital Object Identifier
doi:10.1214/aos/1176325367

Mathematical Reviews number (MathSciNet)
MR1272082

Zentralblatt MATH identifier
0805.62014

JSTOR
links.jstor.org

Subjects
Primary: 62C20: Minimax procedures
Secondary: 62G07: Density estimation 41A25: Rate of convergence, degree of approximation 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Keywords
Bounded normal mean estimation of linear functionals confidence statements for linear functionals modulus of continuity minimax risk nonparametric regression density estimation

Citation

Donoho, David L. Statistical Estimation and Optimal Recovery. Ann. Statist. 22 (1994), no. 1, 238--270. doi:10.1214/aos/1176325367. https://projecteuclid.org/euclid.aos/1176325367


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